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9.35 Use these formulas to implement the hyperbolic sine and hyperbolic cosine functions recursively: sinh2x = 2sinhxcoshx cosh2x = 1 + 2(sinhx)2 sin x x + x 3/6 cos x 1 + x 2/2 Compare your results with the corresponding values of the Math.sinh() and Math.cosh() methods. 9.36 Use these trigonometric formulas to implement the tangent function recursively: tan2 = 2tan /(1 tan2 ) tan x x + x 3/3 Compare your results with the corresponding values of the Math.tan() method. 9.37 Implement a recursive function that evaluates a polynomial a0 + a1 x + a2 x2 + + a3 x3, where the n+1 coefficients ai are passed to the function in an array along with the degree n.
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9.1 The basis of a recursive function is its starting point in its definition and its final step when it is being called recursively; it is what stops the recursion. The recursive part of a recursive function is the assignment that includes the function on the right side of the assignment operator, causing the function to call itself; it is what produces the repetition. For example, in the factorial function, the basis is n! = 1 if n = 0, and the recursive part is n! = n (n 1) if n > 0. The call factorial(10) will generate 10 recursive calls. The call f(6) to the Fibonacci function will generate 14 + 8 = 22 recursive calls because it calls f(5) and f(4), which generate 14 and 8 recursive calls, respectively. A recursive solution is often easier to understand than its equivalent iterative solution. But recursion usually runs more slowly than iteration. Direct recursion is where a function calls itself. Indirect recursion is where a group of functions call each other.
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9.2 9.3 9.4 9.5
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9.1 A recursive function that returns the sum of the first n squares:
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int sum(int n) { if (n == 0) { return 0; } return sum(n-1) + n*n; } // basis // recursion
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A recursive function that returns the sum of the first n powers of a base b:
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double sum(double b, int n) { if (n == 0) { return 1; // basis } return 1 + b*sum(b,n-1); // recursion }
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Note that this solution implements Horner s method: 1 + b*(1 + b*(1 + b*(1 +
+ b))).
RECURSION
A recursive function that returns the sum of the first n elements of an array:
double sum(double[] a, int n) { if (n == 0) { return 0.0; // basis } return sum(a,n-1) + a[n-1]; // recursion }
[CHAP. 9
A recursive function that returns the maximum among the first n elements of an array:
double max(double[] a, int n) { if (n == 1) { return a; // basis } double m = max(a,n-1); // recursion if (a[n-1] > m) { return a[n-1]; } else { return m; } }
A recursive function that returns the maximum among the first n elements of an array and makes no more than lgn recursive calls:
double max(double[] a, int lo, int hi) { if (lo >= hi) { return a[lo]; } int mid = (lo + hi)/2; // middle index double m1 = max(a, lo, mid); // recursion on a[lo..mid] double m2 = max(a, mid + 1, hi); // recursion on a[mid+1..hi] return (m1>m2 m1: m2); // maximum of {m1,m2} }
A recursive function that returns the power xn:
double pow(double x, int n) { if (n == 0) { return 1.0; // basis } return x*pow(x,n-1); // recursion }
A recursive function that returns the power xn and makes no more than lgn recursive calls:
double pow(double x, int if (n == 0) { return 1.0; } double p = pow(x,n/2); if (n%2 == 0) { return p*p; } else { return x*p*p; } } n) { // basis
// recursion (n even) // recursion (n odd)
A recursive function that returns the integer binary logarithm of n: